|ImageIntro=The law of cosines is a trigonometric extension of the Pythagorean Theorem.

|ImageIntro=The law of cosines is a trigonometric extension of the Pythagorean Theorem.

-

|ImageDescElem=

+

|ImageDescElem=The law of cosines is a formula that helps in solving triangles when two or three side lengths of a triangle are known. The formula combines the squares of two side lengths of a triangle and some offset, classified by the cosine of a particular angle, to calculate the square of the third side. For this reason, the law of cosines is often thought of as the generalization of the Pythagorean theorem, which only applies to right triangles. The law of cosines adds an extra term to the Pythagorean theorem so that a third side length of a triangle can be determined when there is no right angle.

-

+

-

+

-

+

-

+

-

+

-

+

-

The law of cosines is a formula that helps in solving triangles when two or three side lengths of a triangle are known. The formula combines the squares of two side lengths of a triangle and some offset, classified by the cosine of a particular angle, to calculate the square of the third side. For this reason, the law of cosines is often thought of as the generalization of the Pythagorean theorem, which only applies to right triangles. The law of cosines adds an extra term to the Pythagorean theorem so that a third side length of a triangle can be determined when there is no right angle.

+

::<math> c^{2} = a^{2} + b^{2} - 2ab \cos C </math>

::<math> c^{2} = a^{2} + b^{2} - 2ab \cos C </math>

When to use it: SAS, SSS.

When to use it: SAS, SSS.

-

+

|ImageDesc===Proof==

-

|ImageDesc=

+

-

+

-

==Proof==

+

===By the Pythagorean Theorem===

===By the Pythagorean Theorem===

An easy way to think of the law of cosines is as an extension of the Pythagorean theorem for a right triangle:

An easy way to think of the law of cosines is as an extension of the Pythagorean theorem for a right triangle:

Contents

Basic Description

The law of cosines is a formula that helps in solving triangles when two or three side lengths of a triangle are known. The formula combines the squares of two side lengths of a triangle and some offset, classified by the cosine of a particular angle, to calculate the square of the third side. For this reason, the law of cosines is often thought of as the generalization of the Pythagorean theorem, which only applies to right triangles. The law of cosines adds an extra term to the Pythagorean theorem so that a third side length of a triangle can be determined when there is no right angle.

When to use it: SAS, SSS.

A More Mathematical Explanation

[Click to view A More Mathematical Explanation]

Proof

By the Pythagorean Theorem

An easy way to think of the law of cosines is as an extens [...]

[Click to hide A More Mathematical Explanation]

Proof

By the Pythagorean Theorem

An easy way to think of the law of cosines is as an extension of the Pythagorean theorem for a right triangle:

By Pythagorean theorem, we know

But is just some portion of side length which is less than the length of . Substituting the difference gives us,

By Pythagorean theorem, we also know that

Substituting the appropriate values gives us,

Expanding the squared term gives us

Simplify for

And by the definition of cosine, we know that

Substituting this value in give us

or

Using the Distance Formula

Let be oriented so that is at the origin, and is at the point.

is the distance from to .

Substituting the appropriate points into the distance formula gives us

Squaring the inner terms, we have

Since ,

Square both sides for

Example Problem

Complete the triangle using the law of cosines.

Solution

[show more][hide]

To find the side length ,

Simplify for

Since , substitution gives us

Simplify for

Taking the square root of both sides gives us

Now we can orient the triangle differently to get get a new version of the law of cosines so we can find angle measure ,